(b) The total number of CheR and CheB molecules are denoted by \(N_R\)  and \(N_B\) , respectively. An unbound CheR molecule resides in the cell cytoplasm and can bind to the receptor dimer if an only if no other enzyme is bound to it. The binding can take place at the tether site or the modification site of the receptor dimer[1]}, [2]}, [3]}. While both these bindings are slow, the tether binding is comparatively faster than the binding at the modification site [4]}, [1]} because of which we only consider tether binding process in our model. The binding takes place with rate \(w_r\)  and once bound the CheR enzyme raises the methylation level of the dimer by one unit with rate \(k_r\) , provided the dimer belongs to an inactive cluster and its methylation level is \(< 8\) . A bound CheR can unbind from the dimer with rate \(w_u\)  and can either reattach to another unoccupied dimer within the same cluster, or return to the cytoplasm. A CheB molecule in the cytoplasm can undergo phosphorylation by an active receptor with rate \(w_p\)  and an unbound CheB-P molecule can dephosphorylate with rate \(w_{dp}\) . The binding, rebinding and unbinding processes for CheB-P are very similar to those for CheR, while demethylation happens if the dimer is active with a non-zero methylation level. The binding between an enzyme molecule and a dimer is a slow process and therefore if one binding event results in only one (de)methylation reaction, then it becomes difficult for the network to maintain perfect adaptation. The possibility of rebinding of the same enzyme molecule to another dimer in the same cluster paves way for multiple (de)methylation of multiple receptors from a single binding event. This is an effective way to include the flavor of assistance neighborhood [6]}, [7]}, [8]} and brachiation [9]} in our model.
